Nonlinear dynamics of a digital phase locked loop

A second-order digital phase-locked loop may exhibit unusual behavior for some parameters due to a fractal boundary between the basin of attraction of the locked fixed point and the attracting basins of coexisting periodic orbits. The usual optimization criterion of the loop parameters using linearized analysis is insufficient, due to coexisting periodic orbits. A new optimization procedure is presented which is based on numerical bifurcation studies. The authors obtain an analytical estimate of average lock time based on the rate of contraction of the phase space in a neighborhood of the fixed point and the size of the phase space that is regular for the underlying Hamiltonian dynamics. The analytical estimate is verified by numerical calculations.>